3.62 \(\int \frac{A+B x^2}{a+b x^2} \, dx\)

Optimal. Leaf size=39 \[ \frac{(A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{B x}{b} \]

[Out]

(B*x)/b + ((A*b - a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(3/2))

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Rubi [A]  time = 0.048688, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{(A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{B x}{b} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(a + b*x^2),x]

[Out]

(B*x)/b + ((A*b - a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(3/2))

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Rubi in Sympy [A]  time = 8.51514, size = 34, normalized size = 0.87 \[ \frac{B x}{b} + \frac{\left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\sqrt{a} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/(b*x**2+a),x)

[Out]

B*x/b + (A*b - B*a)*atan(sqrt(b)*x/sqrt(a))/(sqrt(a)*b**(3/2))

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Mathematica [A]  time = 0.0474816, size = 40, normalized size = 1.03 \[ \frac{B x}{b}-\frac{(a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(a + b*x^2),x]

[Out]

(B*x)/b - ((-(A*b) + a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(3/2))

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Maple [A]  time = 0.003, size = 45, normalized size = 1.2 \[{\frac{Bx}{b}}+{A\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{Ba}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/(b*x^2+a),x)

[Out]

B*x/b+1/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*A-1/b/(a*b)^(1/2)*arctan(x*b/(a*b)^(
1/2))*B*a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235843, size = 1, normalized size = 0.03 \[ \left [\frac{2 \, \sqrt{-a b} B x -{\left (B a - A b\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right )}{2 \, \sqrt{-a b} b}, \frac{\sqrt{a b} B x -{\left (B a - A b\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{\sqrt{a b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(b*x^2 + a),x, algorithm="fricas")

[Out]

[1/2*(2*sqrt(-a*b)*B*x - (B*a - A*b)*log((2*a*b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x
^2 + a)))/(sqrt(-a*b)*b), (sqrt(a*b)*B*x - (B*a - A*b)*arctan(sqrt(a*b)*x/a))/(s
qrt(a*b)*b)]

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Sympy [A]  time = 1.62372, size = 82, normalized size = 2.1 \[ \frac{B x}{b} + \frac{\sqrt{- \frac{1}{a b^{3}}} \left (- A b + B a\right ) \log{\left (- a b \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{2} - \frac{\sqrt{- \frac{1}{a b^{3}}} \left (- A b + B a\right ) \log{\left (a b \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)/(b*x**2+a),x)

[Out]

B*x/b + sqrt(-1/(a*b**3))*(-A*b + B*a)*log(-a*b*sqrt(-1/(a*b**3)) + x)/2 - sqrt(
-1/(a*b**3))*(-A*b + B*a)*log(a*b*sqrt(-1/(a*b**3)) + x)/2

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GIAC/XCAS [A]  time = 0.231714, size = 46, normalized size = 1.18 \[ \frac{B x}{b} - \frac{{\left (B a - A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(b*x^2 + a),x, algorithm="giac")

[Out]

B*x/b - (B*a - A*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b)